Precalculus: Real Mathematics, Real People

901 Pages·2011·175.772 MB·English

by  Ron Larson

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Library of Parent Functions Summary
Linear Function (p. 6) Absolute Value Function (p. 19) Square Root Function (p. 20)
(cid:9)
f(cid:2)x(cid:3)(cid:2)x f(cid:2)x(cid:3)(cid:2)(cid:8)x(cid:8)(cid:2) x, x ≥ 0 f(cid:2)x(cid:3)(cid:2)(cid:7)x
(cid:4)x, x < 0
y y y
2 4
3 f(x) = x
1
f(x) = x f(x) = ⏐x⏐
2
x x
(0, 0) −2 −1 (0, 0) 2 1
−1
x
−1 (0, 0) 2 3 4
−2 −1
Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Domain: (cid:4)0, (cid:3)(cid:3)
Range: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Range: (cid:4)0, (cid:3)(cid:3) Range: (cid:4)0, (cid:3)(cid:3)
Intercept: (cid:2)0, 0(cid:3) Intercept: (cid:2)0, 0(cid:3) Intercept: (cid:2)0, 0(cid:3)
Increasing Decreasing on (cid:2)(cid:4)(cid:3), 0(cid:3) Increasing on (cid:2)0, (cid:3)(cid:3)
Increasing on (cid:2)0, (cid:3)(cid:3)
Even function
y-axis symmetry
Greatest Integer Function (p. 34) Quadratic Function (p. 92) Cubic Function (p. 101)
f(cid:2)x(cid:3)(cid:2)(cid:5)x(cid:6) f(cid:2)x(cid:3)(cid:2) ax2 f(cid:2)x(cid:3)(cid:2) x3
y y y
f(x) = [ [ x]]
3 4 3
2 3 2
1 2
(0, 0)
x 1 f(x) = x2 x
−3 −2 −1 1 2 3 −3 −2 1 2 3
−3 −2 −1 1 2 3 x −1 f(x) = x3
−1 (0, 0) −2
−3 −2 −3
Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3)
Range: the set of integers Range: (cid:4)0, (cid:3)(cid:3) Range: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3)
x-intercepts:in the interval (cid:4)0, 1(cid:3) Intercept: (cid:2)0, 0(cid:3) Intercept: (cid:2)0, 0(cid:3)
y-intercept: (cid:2)0, 0(cid:3) Decreasing on (cid:2)(cid:4)(cid:3), 0(cid:3) Increasing on (cid:2)(cid:4)(cid:3), (cid:3)(cid:3)
Constant between each pair of Increasing on (cid:2)0, (cid:3)(cid:3) Odd function
consecutive integers Even function Origin symmetry
Jumps vertically one unit at Axis of symmetry: x(cid:2) 0
each integer value Relative minimum or vertex: (cid:2)0, 0(cid:3)
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Rational Function (p. 152) Exponential Function (p. 182) Logarithmic Function (p. 195)
1
f(cid:2)x(cid:3)(cid:2) f(cid:2)x(cid:3)(cid:2)ax, a > 0, a(cid:6) 1 f(cid:2)x(cid:3)(cid:2)log x, a > 0, a(cid:6)1
x a
y y y
3 1
2 f(x) = x 1 f(x) = loga x
1 f(x) = ax f(x) = a−x
(0, 1) (1, 0)
x x
−1 1 2 3 1 2
x
−1
Domain: (cid:2)(cid:4)(cid:3), 0(cid:3)(cid:2)(cid:2)0, (cid:3)) Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Domain: (cid:2)0, (cid:3)(cid:3)
Range: (cid:2)(cid:4)(cid:3), 0(cid:3)(cid:2)(cid:2)0, (cid:3)) Range: (cid:2)0, (cid:3)(cid:3) Range: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3)
No intercepts Intercept: (cid:2)0, 1(cid:3) Intercept: (cid:2)1, 0(cid:3)
Decreasing on (cid:2)(cid:4)(cid:3), 0(cid:3)and (cid:2)0, (cid:3)(cid:3) Increasing on (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Increasing on (cid:2)0, (cid:3)(cid:3)
Odd function for f(cid:2)x(cid:3)(cid:2)ax y-axis is a vertical asymptote
Origin symmetry Decreasing on (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Continuous
Vertical asymptote: y-axis for f(cid:2)x(cid:3)(cid:2)a(cid:4)x Reflection of graph of f(cid:2)x(cid:3)(cid:2)ax
Horizontal asymptote: x-axis x-axis is a horizontal asymptote in the line y(cid:2)x
Continuous
Sine Function (p. 293) Cosine Function (p. 293) Tangent Function (p. 304)
f(cid:2)x(cid:3)(cid:2)sin x f(cid:2)x(cid:3)(cid:2)cos x f(cid:2)x(cid:3)(cid:2)tan x
y y
y f(x) = tan x
3 3 3
f(x) = sin x f(x) = cos x
2 2
2
1
1
x x
x
−π π2 π −π −1 π2 π 2π −π π π 3π
2 2 2
−2 −2
−3 −3
(cid:5)
Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Domain:x(cid:6) (cid:7)n(cid:5)
2
Range: (cid:4)(cid:4)1, 1(cid:10) Range: (cid:4)(cid:4)1, 1(cid:10) Range:(cid:2)(cid:4)(cid:3), (cid:3)(cid:3)
Period: 2(cid:5) Period: 2(cid:5)
x-intercepts: (cid:2)n(cid:5), 0(cid:3) (cid:11)(cid:5) (cid:12) Period:(cid:5)
y-intercept: (cid:2)0, 0(cid:3) x-intercepts: 2 (cid:7)n(cid:5), 0 x-intercepts:(cid:2)n(cid:5), 0(cid:3)
y-intercept:(cid:2)0, 0(cid:3)
Odd function y-intercept: (cid:2)0, 1(cid:3)
(cid:5)
Origin symmetry Even function Vertical asymptotes: x(cid:2) (cid:7)n(cid:5)
2
y-axis symmetry
Odd function
Origin symmetry
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Cosecant Function (p. 307) Secant Function (p. 307) Cotangent Function (p. 306)
f(cid:2)x(cid:3)(cid:2)csc x f(cid:2)x(cid:3)(cid:2)sec x f(cid:2)x(cid:3)(cid:2) cot x
1 1 1
y f(x) = csc x = y f(x) = sec x = y f(x) = cot x =
sin x cos x tan x
3 3 3
2 2 2
1 1
x x x
−π π π 2π −π −π π π 3π 2π −π −π π π 2π
2 2 2 2 2 2
−2
−3
(cid:5)
Domain: x(cid:6)n(cid:5) Domain: x(cid:6) (cid:7) n(cid:5) Domain: x(cid:6)n(cid:5)
2
Range: (cid:2)(cid:4)(cid:3), (cid:4)1(cid:10)(cid:2)(cid:4)1, (cid:3)(cid:3) Range: (cid:2)(cid:4)(cid:3), (cid:4)1(cid:10)(cid:2)(cid:4)1, (cid:3)(cid:3) Range: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3)
Period: 2(cid:5) Period: 2(cid:5) Period: (cid:5)
No intercepts (cid:11)(cid:5) (cid:12)
Vertical asymptotes: x(cid:2)n(cid:5) y-intercept: (cid:2)0, 1(cid:3) x-intercepts: 2 (cid:7)n(cid:5), 0
Vertical asymptotes:
Odd function (cid:5) Vertical asymptotes: x(cid:2) n(cid:5)
Origin symmetry x(cid:2) (cid:7)n(cid:5) Odd function
2
Even function Origin symmetry
y-axis symmetry
Inverse Sine Function (p. 319) Inverse Cosine Function (p. 319) Inverse Tangent Function
(p. 319)
f(cid:2)x(cid:3)(cid:2)arcsin x f(cid:2)x(cid:3)(cid:2)arccos x f(cid:2)x(cid:3)(cid:2) arctan x
y y y
π π
π
2 2
f(x) = arccos x
x x
−1 1 −2 −1 1 2
f(x) = arcsin x f(x) = arctan x
π π
− x −
2 −1 1 2
Domain: (cid:4)(cid:4)1, 1(cid:10) Domain: (cid:4)(cid:4)1, 1(cid:10) Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3)
Range: (cid:13)(cid:4)(cid:5), (cid:5)(cid:14) Range: (cid:4)0, (cid:5)(cid:10) Range: (cid:11)(cid:4)(cid:5), (cid:5)(cid:12)
2 2 (cid:11) (cid:5)(cid:12)
2 2
Intercept: (cid:2)0, 0(cid:3) y-intercept: 0, 2 Intercept: (cid:2)0, 0(cid:3)
Odd function (cid:5)
Horizontal asymptotes: y(cid:2)±
Origin symmetry 2
Odd function
Origin symmetry
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
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1111427631 Precalc SE FM.qxd 10/12/10 3:49 PM Page i
Precalculus
Real Mathematics, Real People
Sixth Edition
Ron Larson
The Pennsylvania State University, The Behrend College
With the assistance of David C. Falvo
The Pennsylvania State University, The Behrend College
y
y
3
8

2
2x x2 3x 8
6
f x
f x
x 2
x 3
4
x
−
2 1 2 3
2 −
1
− x 2 4
x 2
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
4 6 8 − x 2 1
3
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1111427631 Precalc SE FM.qxd 10/12/10 3:49 PM Page ii
Precalculus: Real Mathematics, Real People © 2012, 2008 Brooks/Cole, Cengage Learning
Sixth Edition
ALL RIGHTS RESERVED. No part of this work covered by the copyright
Ron Larson
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1111427631 Precalc SE FM.qxd 10/12/10 3:49 PM Page iii
Precalculus
Real Mathematics, Real People
Sixth Edition
Contents
Chapter 1 Functions and Their Graphs
1
Introduction to Library of Functions 2
1.1 Lines in the Plane 3
1.2 Functions 16
1.3 Graphs of Functions 29
1.4 Shifting, Reflecting, and Stretching Graphs 41
1.5 Combinations of Functions 50
1.6 Inverse Functions 60
1.7 Linear Models and Scatter Plots 71
Chapter Summary 80 Review Exercises 82
Chapter Test 86 Proofs in Mathematics 87
Chapter 2 Polynomial and Rational Functions
89
2.1 Quadratic Functions 90
2.2 Polynomial Functions of Higher Degree 100
2.3 Real Zeros of Polynomial Functions 113
2.4 Complex Numbers 128
2.5 The Fundamental Theorem of Algebra 135
2.6 Rational Functions and Asymptotes 142
y
y 2.7 Graphs of Rational Functions 151
2.8 Quadratic Models 161
Chapter Summary 168 Review Exercises 170
Chapter Test 175 Proofs in Mathematics 176
8
Progressive Summary (Chapters 1 and 2) 178
L
1
L
Chapter 3 Exponential and Logarithmic Functions
179
3
6
3.1 Exponential Functions and Their Graphs 180
3.2 Logarithmic Functions and Their Graphs 192
3.3 Properties of Logarithms 203
4 3.4 Solving Exponential and Logarithmic Equations 210
x
3.5 Exponential and Logarithmic Models 221
3.6 Nonlinear Models 233
L
2 Chapter Summary 242 Review Exercises 244
2
x
Chapter Test 248 Cumulative Test: Chapters 1–3 249
Proofs in Mathematics 251
Progressive Summary (Chapters 1–3) 252
x
iii
4 6 8
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
1111427631 Precalc SE FM.qxd 10/12/10 3:49 PM Page iv
iv Contents
Chapter 4 Trigonometric Functions
253
4.1 Radian and Degree Measure 254
4.2 Trigonometric Functions: The Unit Circle 265
4.3 Right Triangle Trigonometry 273
4.4 Trigonometric Functions of Any Angle 284
4.5 Graphs of Sine and Cosine Functions 292
4.6 Graphs of Other Trigonometric Functions 304
4.7 Inverse Trigonometric Functions 315
4.8 Applications and Models 326
Chapter Summary 338 Review Exercises 340
Chapter Test 345 Library of Parent Functions Review 346
Proofs in Mathematics 348
Chapter 5 Analytic Trigonometry
349
5.1 Using Fundamental Identities 350
5.2 Verifying Trigonometric Identities 357
5.3 Solving Trigonometric Equations 365
5.4 Sum and Difference Formulas 377
5.5 Multiple-Angle and Product-to-Sum Formulas 384
Chapter Summary 394 Review Exercises 396
Chapter Test 399 Proofs in Mathematics 400
Chapter 6 Additional Topics in Trigonometry
403
6.1 Law of Sines 404
6.2 Law of Cosines 413
6.3 Vectors in the Plane 420
6.4 Vectors and Dot Products 434
6.5 Trigonometric Form of a Complex Number 443
Chapter Summary 456 Review Exercises 458 y
Chapter Test 461 Cumulative Test: Chapters 4–6 462
Proofs in Mathematics 464
Progressive Summary (Chapters 1–6) 468
3

2
x
−
2 1 2 3
2x −
1
f x
x 3 −
2
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
1111427631 Precalc SE FM.qxd 10/12/10 3:49 PM Page v
Contents v
Chapter 7 Linear Systems and Matrices
469
7.1 Solving Systems of Equations 470
7.2 Systems of Linear Equations in Two Variables 480
7.3 Multivariable Linear Systems 489
7.4 Matrices and Systems of Equations 504
7.5 Operations with Matrices 518
7.6 The Inverse of a Square Matrix 532
7.7 The Determinant of a Square Matrix 541
7.8 Applications of Matrices and Determinants 548
Chapter Summary 558 Review Exercises 560
Chapter Test 566 Proofs in Mathematics 567
Chapter 8 Sequences, Series, and Probability
569
8.1 Sequences and Series 570
8.2 Arithmetic Sequences and Partial Sums 581
8.3 Geometric Sequences and Series 589
8.4 The Binomial Theorem 599
8.5 Counting Principles 607
8.6 Probability 616
Chapter Summary 626 Review Exercises 628
Chapter Test 631 Proofs in Mathematics 632
Chapter 9 Topics in Analytic Geometry
635
9.1 Conics: Circles and Parabolas 636
9.2 Ellipses 647
9.3 Hyperbolas and Rotation of Conics 656
y
9.4 Parametric Equations 669
y
9.5 Polar Coordinates 677
9.6 Graphs of Polar Equations 683
9.7 Polar Equations of Conics 691
8 Chapter Summary 698 Review Exercises 700
L
Chapter Test 704 Cumulative Test: Chapters 7–9 705
1
Proofs in Mathematics 707 L
3
Progressive Summary (Chapters 3–9) 710
6

Chapter 10 Analytic Geometry in Three Dimensions
711
10.1 The Three-Dimensional Coordinate System 712
4
10.2 Vectors in Space 719 x
10.3 The Cross Product of Two Vectors 726
10.4 Lines and Planes in Space 733 L
2
2
x Chapter Summary 742 Review Exercises 744
Chapter Test 746 Proofs in Mathematics 747
x
4 6 8
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.